These classes are derived from the class Model, each implementing
a different reproduction kernel for the Hawkes process.
They inherit all fields from Model.
The kernel Exponential has density function
$$h^\ast(t) = \beta \exp(-\beta t) 1_{\{t \ge 0\}}.$$
Its vector of parameters must be of the form \((\eta, \mu, \beta)\).
Both loglik, its derivatives, and whittle can be used with this reproduction kernel.
The kernel SymmetricExponential has density function
$$h^\ast(t) = 0.5 \beta \exp(-\beta |t|).$$
Its vector of parameters must be of the form \((\eta, \mu, \beta)\).
Only whittle can be used with this reproduction kernel.
The kernel Gaussian has density function
$$h^\ast(t) = \frac{1}{\sigma \sqrt{2\pi}}\exp\left(-\frac{(t-\nu)^2}{2\sigma^2}\right).$$
Its vector of parameters must be of the form \((\eta, \mu, \nu, \sigma^2)\).
Only whittle is available with this reproduction kernel.
The kernel PowerLaw has density function
$$h^\ast(t) = \theta a^\theta (t+a)^{-\theta-1} 1_{\{\theta > 0 \}}.$$
Its vector of parameters must be of the form \((\eta, \mu, \theta, a)\).
Both loglik, its derivatives, and whittle can be used with this reproduction kernel.
The kernels Pareto3, Pareto2 and Pareto1 have density function
$$h_\theta^\ast(t) = \theta a^\theta t^{-\theta - 1} 1_{\{t > a\}},$$
with \(\theta\) = 3, 2 and 1 respectively.
Their vectors of parameters must be of the form \((\eta, \mu, a)\).
Only whittle is available with this reproduction kernel.
Model